Intrinsic ergodicity beyond specification: beta-shifts, S-gap shifts, and their factors

Abstract

We give sufficient conditions for a shift space $(\Sigma ,\sigma )$ to be intrinsically ergodic, along with sufficient conditions for every subshift factor of $\Sigma$ to be intrinsically ergodic. As an application, we show that every subshift factor of a $\beta$-shift is intrinsically ergodic, which answers an open question included in Mike Boyle's article "Open problems in symbolic dynamics". We obtain the same result for $S$-gap shifts, and describe an application of our conditions to more general coded systems. One novelty of our approach is the introduction of a new version of the specification property that is well adapted to the study of symbolic spaces with a non-uniform structure.

27 pages, 1 figure, minor changes based on referee's suggestions, in v.3 the result on coded systems has been moved to the introduction, resulting in small changes to numbering (lettering) of results

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